Laws of form

The remarkable book Laws of Form by George Spencer-Brown was published in 1969 and is almost forgotten today. The best expositors have been William Bricken with his boundary mathematics, Louis Kauffman with his knot theory, and Francisco Varela with his work on self-reference. Otherwise it has become something of an underground classic but otherwise forgotten. There are several reasons for the latter, including the exaggerated claims of the author and some enthusiasts. That said, I think it’s worth rehabilitating the Laws of Form (LoF) and rightly discerning its significance.

LoF is a work on diagrammatic reasoning in the tradition of Leibniz and CS Peirce. It is a calculus, complete with arithmetic and algebra, based on the act of making and indicating a distinction. Thus it is a work of mathematical realism, which begins to explain why it is not of interest to anti-realists. Its greatest accomplishment is the unified treatment of injunction and indication, of implication and negation via a single symbol, called a cross.

Here are the arithmetic axioms of the calculus of indications:

That’s it. The inverted “L” is the cross symbol. A cross next to another cross is equal to one cross; this is the Law of Calling. A cross inside another cross is equal to blank, that is, as if no cross had been written. This is the Law of Crossing, hence the name of the symbol, Cross.

This is a two-dimensional calculus, which gives it advantages that one-dimensional notation does not have. It also makes it hard to display typographically. The best alternative is simply to use parentheses or brackets:

( ) ( ) = ( ) and (( )) = .

These arithmetic axioms can be used to derive two algebraic axioms:

((A) (B)) C = ((A C) (B C)) and ((A) A) = .

From this a complete calculus can be constructed. It is isomorphic to Boolean algebra and other functionally-complete binary calculi, which is another reason LoF hasn’t stirred a lot of interest.

Things get more interesting as we review where this calculus comes from. Again this exhibits its realism; the standard approach for mathematics and symbolic logic is to begin with algebraic axioms or postulates without reference to any model or reality.

Let’s begin with a blank surface, say a blank page of paper. Now draw a distinction on this surface; that is, draw a closed curve or divide the page into two parts. Notice what has happened: part of the paper is distinguished from the rest of the paper by being to one side of the curve, say the inside. The curve separates the other side from the inside; call it the outside. But the original piece of paper is still there. We can still consider the whole piece of paper.

This process is symbolized by LoF as follows: what is outside the cross (or parentheses) can be seen inside the cross (or parentheses) if we change perspectives to the whole page. This is symbolized in a theorem:

(A) B = (A B) B.

So the distinction that is drawn is not between two contraries but within one space, represented by the whole page. It also shows the distinction can be undermined. This has been exploited to represent self-reference.